Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

摘要

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups.View full textDownload full textKey WordsEndomorphisms of groups, 2-Engel Groups, p-Groups, Near-rings2000 Mathematics Subject Classification20D45, 20E36Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00927870801941903